I have a question about what defines “nested models” that can be compared using the likelihood ratio test (LRT). For example, given the 3 linear models…

M1: Y = Intercept + B1 * X1 + B2 * X2

M2: Y = Intercept + B1 * X1

M3: Y = Intercept + B3 * (0.5 * X1 + 0.5 * X2)

M1 and M2 are nested models because M2 is a special case of M1 with B2 = 0. M1 and M2 can be compared using the LRT with 1 degree of freedom.

The model M3 might arise if you decided to model Y as a function of a pre-specified linear combination of X1 and X2 (in this example I’ve used the mean of X1 and X2). Are M1 and M3 nested models? My thinking is that M3 is a special case of M1 with B1 = B3 * 0.5 and B2 = B3 * 0.5. Can M1 and M3 be compared using the LRT? With 1 degree of freedom?

Many thanks

Stephen